Finite Group Databasebeta

← #10691479

Subgroups — #10691479

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79297C211yesmaximal, characteristic
2#73299C211yesmaximal, characteristic, complemented
3#70789C211yesmaximal, characteristic, complemented
4#70762C211yesmaximal, characteristic, complemented
5#69395C211yesmaximal, characteristic, complemented
6C3×#1276C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#9237—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×#872—21no—
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×C43C2211yescharacteristic
15#4940—22no—
16#4939—21no—
17C4×(C2×D3)⋊C4—21no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4928—21no—
20C2×#805—21no—
21C2×#805—21no—
22#4924C2211yescharacteristic
23#4846C412yescomplemented
24#4843C412yescomplemented
25#4736—21no—
26#4732—21no—
27C4×D24—22no—
28#4470—22no—
29#4459—22no—
30#3100—31no—
31#2293—31no—
32#1962—31no—
33#1502—31no—
34#1486—31no—
35#1303—31no—
36#1276D311yescharacteristic, complemented
37C2×#235—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C2×C3×C42—21no—
41C2×C3×C42—21no—
42C2×C3×C42D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×C3×C42C2311yescharacteristic
45#930—21no—
46#930—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×Dic3⋊C4—22no—
53C2×C4×Dic3—21no—
54C2×#208—21no—
55C2×D24—22no—
56#875—21no—
57#874C2×C411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12C2×C412yes—
63C4×D12D412yes—
64#872—42no—
65#872—42no—
66#869C2×C411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yes—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#821—22no—
76#821—22no—
77#818—22no—
78#805—42no—
79#805—42no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82#532—31no—
83C2×C4×D4—31no—
84C2×#117—61no—
85#480—61no—
86#474—32no—
87#459—32no—
88#447—61no—
89C4×D8—62no—
90#431C2×D311yescharacteristic
91#431C2×D311yescharacteristic
92#423—31no—
93C2×#104—61no—
94C2×#104—61no—
95#390—61no—
96#389—61no—
97#388—62no—
98C4×C4⋊C4—31no—
99C4×C22⋊C4—61no—
100C43C2×D311yescharacteristic
101#327—62no—
102D4⋊C8—62no—
103C23×D3—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C2×#67—41no—
110C22×Dic3—21no—
111C22×Dic3—22no—
112#235—41no—
113#235—41no—
114#235—41no—
115#235—41no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—42no—
121C2×D12C22⋊C412yes—
122C2×C4×D3—22no—
123C2×C4×D3—22no—
124C2×C4×D3—41no—
125C2×C4×D3—41no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C42—21no—
133C3×C42—21no—
134C3×C42—21no—
135C3×C42—42no—
136C3×C42—42no—
137C3×C42D812yescomplemented
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—22no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—44no—
145(C2×D3)⋊C4—44no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—42no—
152C4⋊Dic3C22⋊C412yes—
153Dic3⋊C4—44no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156#208—41no—
157#208—41no—
158C2×C3⋊C8—41no—
159D24—44no—
160C2×#41—61no—
161C22×D4—31no—
162C23×C4—61no—
163C2×D8—62no—
164C2×M16—61no—
165C4⋊Q8—31no—
166#127—31no—
167#124—61no—
168#121—61no—
169#121—61no—
170C4×D4—32no—
171C4×D4—32no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176#117—122no—
177#117—122no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—61no—
187C2×C42D1211yescharacteristic
188C2×C42D1211yescharacteristic
189C2×C42C22×D311yescharacteristic
190#107—62no—
191C4⋊C8—21no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8C4×D312yes—
195#104—122no—
196#104—122no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203#95—32no—
204#95—62no—
205C23×C3#12011yescharacteristic
206C22×D3—22no—
207C22×D3—42no—
208C22×D3—42no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4C2×C22⋊C411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12311yescharacteristic
226C2×C3×C4C2×D811yescharacteristic
227C2×C3×C4#13511yescharacteristic
228C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230#67—81no—
231#67—81no—
232C2×Dic3—22no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239D12—22no—
240D12—22no—
241D12—42no—
242D12—42no—
243D12—41no—
244D12—41no—
245D12—81no—
246C4×D3—42no—
247C4×D3—42no—
248C4×D3—42no—
249C4×D3—42no—
250C4×D3—81no—
251C4×D3—81no—
252Dic6—41no—
253Dic6—41no—
254C3×C8—42no—
255C3×C8—81no—
256C3⋊C8—81no—
257C24—61no—
258#41—121no—
259#41—121no—
260#41—121no—
261#41—121no—
262C2×Q8—61no—
263C2×D4—32no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—122no—
269C2×D4—121no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—61no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—121no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4C2×D1211yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284D8—124no—
285M16—121no—
286M16—121no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—121no—
290C4⋊C4—32no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—124no—
295C4⋊C4—122no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—124no—
299C22⋊C4—124no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—42no—
306C42—42no—
307C42—121no—
308C42—121no—
309C42D2412yescomplemented
310C42C2×C4×D311yescharacteristic
311C42#23511yescharacteristic
312C42D3×D411yescharacteristic
313C42#23911yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3#38611yescharacteristic
317C22×C3#44711yescharacteristic
318C22×C3#44911yescharacteristic
319C2×D3—42no—
320C2×D3—42no—
321C2×D3—42no—
322C2×D3—82no—
323C2×D3—82no—
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—42no—
331C3×C4—42no—
332C3×C4#42112yes—
333C3×C4C4×D811yescharacteristic
334C3×C4#44211yescharacteristic
335C3×C4#45911yescharacteristic
336C3×C4#46311yescharacteristic
337Dic3—42no—
338Dic3—82no—
339Dic3—81no—
340C23—62no—
341C23—122no—
342C23—122no—
343C23—121no—
344C23—121no—
345C23—121no—
346C23#88211yescharacteristic
347Q8—121no—
348Q8—121no—
349D4—62no—
350D4—62no—
351D4—122no—
352D4—121no—
353D4—121no—
354D4—122no—
355D4—241no—
356D4—241no—
357D4—241no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—62no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—121no—
376C2×C4—122no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4C4×D1211yescharacteristic
380C2×C4D3×C22⋊C411yescharacteristic
381C2×C4#88411yescharacteristic
382C2×C4#89311yescharacteristic
383C2×C4#89411yescharacteristic
384C2×C4#89511yescharacteristic
385C2×C4C2×D2411yescharacteristic
386C2×C4#90411yescharacteristic
387C2×C4#90811yescharacteristic
388C2×C4#90811yescharacteristic
389C8—42no—
390C8—81no—
391C8—241no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3#187911yescharacteristic
395C2×C3#189711yescharacteristic
396C2×C3#190611yescharacteristic
397D3—82no—
398D3—161no—
399C22—21no—
400C22—21no—
401C22—122no—
402C22—122no—
403C22—122no—
404C22—242no—
405C22—242no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22#466511yescharacteristic
410C22#473211yescharacteristic
411C22#473411yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—42no—
417C4—122no—
418C4—242no—
419C4—241no—
420C4C4×D2411yescharacteristic
421C4#471111yescharacteristic
422C4#482112yes—
423C4#484311yescharacteristic
424C4#484411yescharacteristic
425C3#923711yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—242no—
429C2—481no—
430C2#6867311yescharacteristic
431C2#6875711yescharacteristic
432C2#6877811yescharacteristic
subgroupG/H
centerC2×C4#895
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#386
prime radical#79297C2
cosocleC2×C3×C42C23
hypercenter#1276D3
hypocenterC3#9237
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #73299C2semidirect
3. #70789C2semidirect
4. #70762C2semidirect
5. #69395C2semidirect
6. C3×#1276C2semidirect
7. #67746C2semidirect
23. #4846C4semidirect
24. #4843C4semidirect
36. #1276D3semidirect
137. C3×C42D8semidirect
309. C42D24semidirect
425. C3#9237semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#792972
#732992
#707892
#707622
#693952
C3×#12762
#677462
#92373